Algebraic Geometry and Arithmetic Curves

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9780199202492: Algebraic Geometry and Arithmetic Curves

This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.

The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group.

The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford.

This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students.

Le informazioni nella sezione "Riassunto" possono far riferimento a edizioni diverse di questo titolo.

Recensione:

Review from previous edition Will be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field. ( EMS)

This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. ( Zentralblatt MATH)

A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. ( Zentralblatt MATH)

L'autore:

Qing Liu is Charge de recherche, Centre National de la Recherche Scientifique (CNRS) Laboratoire de Theorie des Nombres et d'Algorithmique Arithmetique, Universite Bordeaux 1.

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Centre National de La Recherche Scientifique (Cnrs) Laboratoire de Theorie Des Nombres Et D Algorith
Editore: Oxford University Press, United Kingdom (2006)
ISBN 10: 0199202494 ISBN 13: 9780199202492
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Descrizione libro Oxford University Press, United Kingdom, 2006. Paperback. Condizione libro: New. 234 x 155 mm. Language: English . Brand New Book. This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski s Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck s duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo s criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students. Codice libro della libreria AAZ9780199202492

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Qing Liu
Editore: Oxford University Press, United Kingdom (2006)
ISBN 10: 0199202494 ISBN 13: 9780199202492
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Descrizione libro Oxford University Press, United Kingdom, 2006. Paperback. Condizione libro: New. 234 x 155 mm. Language: English . Brand New Book. This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski s Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck s duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo s criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students. Codice libro della libreria AAZ9780199202492

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Descrizione libro Condizione libro: New. Depending on your location, this item may ship from the US or UK. Codice libro della libreria 97801992024920000000

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Descrizione libro OUP Oxford 2006-06-29, 2006. Condizione libro: New. Brand new book, sourced directly from publisher. Dispatch time is 24-48 hours from our warehouse. Book will be sent in robust, secure packaging to ensure it reaches you securely. Codice libro della libreria NU-GRD-04958955

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Descrizione libro Oxford University Press 2006-06-29, Oxford, 2006. paperback. Condizione libro: New. Codice libro della libreria 9780199202492

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Descrizione libro OUP Oxford, 2006. PAP. Condizione libro: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Codice libro della libreria GB-9780199202492

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Descrizione libro Oxford University Press. Paperback. Condizione libro: new. BRAND NEW, Algebraic Geometry and Arithmetic Curves, Qing Liu, Reinie Erne, This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students. Codice libro della libreria B9780199202492

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Descrizione libro OUP Oxford, 2006. Condizione libro: New. book. Codice libro della libreria ria9780199202492_rkm

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